Tighter Theory for Local SGD on Identical and Heterogeneous Data (1909.04746v4)

Abstract: We provide a new analysis of local SGD, removing unnecessary assumptions and elaborating on the difference between two data regimes: identical and heterogeneous. In both cases, we improve the existing theory and provide values of the optimal stepsize and optimal number of local iterations. Our bounds are based on a new notion of variance that is specific to local SGD methods with different data. The tightness of our results is guaranteed by recovering known statements when we plug $H=1$, where $H$ is the number of local steps. The empirical evidence further validates the severe impact of data heterogeneity on the performance of local SGD.

Better Analysis for Local SGD for Identical and Heterogeneous Data

The paper "Better Analysis for Local SGD for Identical and Heterogeneous Data" explores the theoretical underpinnings of Local Stochastic Gradient Descent (Local SGD), a widely employed optimization method in distributed machine learning. Local SGD is particularly relevant in federated learning and parallel computing, where communication costs are a critical concern. This paper focuses on two scenarios: when data is identical across nodes (i.e., IID data) and when data is heterogeneous (non-IID).

Theoretical Advancements

The authors present a comprehensive theoretical analysis of Local SGD, improving upon existing results in several ways. Their primary objective is to derive improved convergence rates and remove several restrictive assumptions that have been prevalent in prior analyses. Specifically, they relax the bounded variance assumption in the IID setting and address the bounded dissimilarity and bounded gradients assumptions in the non-IID scenario.

1. Local SGD with IID Data:
   • For IID data, the paper shows that Local SGD can achieve the same convergence rate as Minibatch SGD while significantly reducing communication overhead. By carefully choosing the synchronization interval $H$, the authors demonstrate that one can maintain the same asymptotic convergence rate of Minibatch SGD (i.e., $1/(MT)$) up to logarithmic factors and constants.
2. Local SGD with Heterogeneous Data:
   • In the more challenging non-IID case, the authors contribute novel convergence bounds without assuming bounded dissimilarity or gradients. They introduce a variance measure $\sigma_f^2$, which provides a meaningful characterization of the variance in Local SGD. Their analysis captures the true data heterogeneity across different nodes, which is a significant departure from traditional assumptions.

Contributions and Implications

The paper's contributions lie in both theoretical and practical domains. From a theoretical perspective, the results extend the applicability of Local SGD to broader settings without the stringent assumptions that have been a haLLMark of earlier work. Practically, these findings are crucial for federated learning applications where data distributions are inherently non-IID and communication efficiency is paramount.

The paper also discusses the implications of their theoretical results through extensive experiments. These experiments confirm the robustness of their analysis across different datasets and heterogeneous settings, showcasing the practical value of their theoretical insights.

Future Directions

This research opens multiple avenues for future investigations. Firstly, extending these findings to more complex models and scenarios, such as adversarial settings or privacy-preserving decentralized learning, would be beneficial. Additionally, exploring the integration of these improved Local SGD methods with other optimization techniques could yield further enhancements in efficiency and performance.

In summary, the paper provides significant advancements in the analysis of Local SGD, both in the presence of identical and diverse data distributions. The results bolster our understanding of Local SGD’s efficiency and introduce more adaptable theoretical frameworks that negate the need for restrictive assumptions, making it a vital reference for researchers exploring distributed optimization in machine learning.